Collaborative Research: Ergodic Control of Stochastic Differential Equations Driven By a Class of Pure-Jump Levy Processes, and Applications to Stochastic Networks

大规模网络中的排队过程在社会中无处不在，医院、电话呼叫中心、服务网络以及制造和服务运营管理中的患者流就是例证。这项研究将为批次到达情况下的排队过程提供基本的理论理解。这项研究的结果将通过与卫生系统和系统分析行业的合作，直接应用于现实世界的随机网络。这项研究将支持代表不足的少数群体，并培训年轻的STEM毕业生掌握新的数学技能。研究了由纯跳跃Levy过程驱动的Ito随机微分方程(SDE)描述的系统的遍历控制问题。本文的研究目标是：(1)建立一大类纯跳跃Levy过程驱动的受控随机微分方程的遍历控制的综合理论框架；(2)研究接纳控制中一个新的完全非线性问题，该问题不属于通常的随机控制范式；(3)发展数值迭代算法和空间逼近方法，以研究大时间渐近性，并促进相关的Hamilton-Jacobi-Bellman(HJB)方程的数值解，以及在存在未知参数的情况下设计自适应控制的学习方案。这项研究将推进应用数学和随机控制的基础科学，为应用概率和随机网络的研究做出基础性贡献。

Queueing processes in large-scale networks are ubiquitous in society as exemplified by patient flow in hospitals, telephone call centers, service networks, and manufacturing and service operations management. This research will provide fundamental theoretical understanding of these queueing processes in the presence of batch arrivals. The results of this research will be put into direct applications to real-world stochastic networks through collaborations with the health systems and system analysis industry. This research will support under-represented minority groups and train young STEM graduates with new mathematical skills. This research concerns ergodic control problems for systems described by Ito stochastic differential equations (SDEs) driven by pure-jump Levy processes. The research objectives are: (1) to develop a comprehensive theoretical framework for ergodic control for a large class of controlled SDEs driven by a pure-jump Levy process; (2) to study a novel fully nonlinear problem that arises in admission control and falls outside the usual paradigm of stochastic control; and (3) to develop value iteration algorithms and spatial approximation methods in order to study large time asymptotics, and also to facilitate the numerical solution of the associated Hamilton-Jacobi-Bellman (HJB) equations, and the design of learning schemes for adaptive control in the presence of unknown parameters. This research will advance the basic science of applied mathematics and stochastic control, and make fundamental contributions to applied probability and stochastic networks.